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On Geodesic Homotopies of Controlled Width and Conjugacies in Isometry Groups

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On Geodesic Homotopies of Controlled Width and Conjugacies in Isometry Groups Groups Geom. Dyn. 7 (2013), 911–929 Groups, Geometry, and Dynamics DOI 10.4171/GGD/210 © European Mathematical Society On geodesic homotopies of controlled width and conjugacies in isometry groups Gerasim Kokarev Abstract. We give an analytical proof of the Poincaré-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space. Mathematics Subject Classification (2010). 53C23, 58E20, 20F65. Keywords. Homotopy width, harmonic maps, Hadamard space, decision problems. 1. Introduction Let M and M 0 be smooth Riemannian manifolds without boundary. For a smooth mapping uW M ! M 0 by E.u/ we denote its energy Z E.u/ D kdu.x/k2 dvol.x/; M 0 where the norm of the linear operator du.x/W TxM ! Tu.x/M is induced by the Riemannian metrics on M and M 0. Let u and v be smooth homotopic mappings of 0 M to M and H.s; / be a smooth homotopy between them. The L2-width W2.H / of H is defined as the L2-norm of the function `H .x/ D the length of the curve s 7! H.s;x/; x 2 M: (1.1) A smooth homotopy H.s;x/ is called geodesic if for each x 2 M the track curve s 7! H.s;x/ is a geodesic. In [7], [8] Kappeler, Kuksin, and Schroeder prove the following geometric in- 0 equality for the L2-widths of geodesic homotopies when the target manifold M is non-positively curved. Width Inequality I. Let M and M 0 be compact Riemannian manifolds and suppose that M 0 has non-positive sectional curvature. Let be a homotopy class of maps 912 G. Kokarev 0 of M to M . Then there exist constants C and C with the following property: any smooth homotopic maps u and v 2 can be joined by a geodesic homotopy H whose L2-width W2.H / is controlled by the energies of u and v, 1=2 1=2 W2.H / 6 C.E .u/ C E .v// C C: (1.2) 0 Moreover, if the sectional curvature of M is strictly negative, the constants C and C can be chosen to be independent of a homotopy class . This inequality can be viewed as a version of the Poincaré inequality for mappings between manifolds. It also has an isoperimetric flavour; it says that the ‘measure’ of the cylinder induced by the homotopy is estimated in terms of the ‘measure’ of its boundary. Inequality (1.2) is a key ingredient in the proof of compactness results for perturbed harmonic map equation [7], [9]. The latter, combined with old results of Uhlenbeck, yields Morse inequalities for harmonic maps with potential [10]. The proof of Width Inequality I in [7], [8] is based on an analogous inequality for maps of metric graphs; see [7], Th. 5.1. In more detail, let G be a finite graph and uW G ! M 0 be a smooth map, that is, whose restriction to every edge is smooth. The length L.u/ of u is defined as the sum of the lengths of the images of the edges. By the L1-width W1.H / of a homotopy H we mean the L1-norm of the length function `H .x/, given by (1.1). Width Inequality II. Let G be a finite graph and M 0 be a compact manifold of non-positive sectional curvature. Let be a homotopy class of maps G ! M 0. Then there exist constants C and C with the following property: any smooth homotopic maps u and v 2 can be joined by a geodesic homotopy H such that W1.H / 6 C.L.u/ C L.v// C C: 0 Moreover, if the sectional curvature of M is strictly negative, the constants C and C can be chosen to be independent of a homotopy class . The purpose of this note is twofold: firstly, we generalise the width inequalities to the framework of equivariant maps, which we assume to be valued in Hadamard metric spaces. This, in particular, includes width inequalities for homotopies between maps into non-compact target spaces, completing the previous results [8] in this direction. In contrast with the geometric methods in [7], [8] (and also algebraic in [2]), we give an analytical proof of the width inequalities via harmonic map theory. Secondly, we use width inequalities for equivariant maps of trees to study the multiple conjugacy problem for finitely generated groups ƒ acting by isometries on Hadamard spaces. More precisely, under some extra hypotheses, we obtain a linear estimate for the conjugator length in these groups: given two finite conjugate lists of elements .ai /16i6N and .bi /16i6N in ƒ there exists a conjugator g 2 ƒ, On geodesic homotopies of controlled width and conjugacies in isometry groups 913 1 bi D g ai g such that XN jgj 6 C .jai jCjbi j/ C C; iD1 where jj stands for the length d. ;e/in the word metric on ƒ. If the group ƒ has a soluble word problem, then the latter estimate yields immediately the solubility of the multiple conjugacy problem in ƒ. Acknowledgments. The author is grateful to Sergei Kuksin and Dieter Kotschick for their interest in the work and a number of discussions on the subject. The paper was written when the author held a personal EPSRC fellowship at the University of Edinburgh. 2. Statements and discussion of results 2.1. Width inequalities for equivariant maps. Let M be a compact Riemannian manifold without boundary; we denote by Mz its universal cover and by the fun- damental group 1.M /. Let .Y; d/ be an Hadamard space; that is a complete length space of non-positive curvature in the sense of Alexandrov (see Section 3 for a precise definition). Denote by a representation of in the isometry group of Y . Recall that a map uW Mz ! Y is called -equivariant if u.g x/ D .g/ u.x/ for all x 2 M;z g 2 : For -equivariant maps u and v the real-valued functions d.u.x/; v.x//, where x 2 Mz , are invariant with respect to the domain action and, hence, are defined on the quotient M D M=z . In particular, the quantity  Z Ã1=2 2 d2.u; v/ D d .u.x/; v.x// dvol.x/ (2.1) M defines a metric on the space of locally L2-integrable -equivariant maps. The latter can be also regarded as the L2-width of a unique geodesic homotopy between - equivariant maps. If u is a locally Sobolev W 1;2-smooth -equivariant map, then its energy density measure jduj2 dvol (see Section 3) is also -invariant and the energy of u is defined as the integral Z E.u/ D jduj2 dvol : M Recall that the ideal boundary of Y is defined as the set of equivalence classes of asymptotic geodesic rays; two rays are asymptotic if they remain at a bounded distance from each other. Clearly, any action of by isometries on Y extends to the action on the ideal boundary. 914 G. Kokarev Theorem 1. Let M be a compact Riemannian manifold without boundary and Y be a locally compact Hadamard space. Let be the fundamental group of M and W ! Isom.Y / be its representation whose image does not fix a point on the ideal boundary of Y . Then there exists a constant C such that for any -equivariant 1;2 locally W -smooth maps u and v the L2-width of a geodesic homotopy H between them satisfies the inequality 1=2 1=2 W2.H / 6 C.E .u/ C E .v//: (2.2) The proof appears in Section 4. The idea is to prove first a similar inequality when one of the maps is an energy minimiser, and then to use compactness properties of the moduli space formed by such minimisers. The former is based on a compactness argument, mimicking the proof of the classical Poincaré inequality. Below we state a version of Theorem 1 for equivariant maps of trees. First, we introduce more notation. Let G be a finite connected graph without terminals and be its fundamental group 1.G/.ByT we denote the universal covering tree of G; the group acts naturally on T by the deck transformations. As above the symbol denotes a representation of in the isometry group of an Hadamard space Y . For a locally rectifiable -equivariant map uW T ! Y , its length density measure jdujdt (see Section 3)is-invariant and the length of u is defined as the integral Z L.u/ D jdujdt: G Theorem 2. Let G be a finite graph and Y be a locally compact Hadamard space. Let be the fundamental group of G and W ! Isom.Y / be its representation whose image does not fix a point on the ideal boundary of Y . Then there exists a constant C such that for any locally rectifiable -equivariant maps u and v the L1-width of a geodesic homotopy between them satisfies the inequality W1.H / 6 C.L.u/ C L.v//: Example. Let M 0 be a non-compact Riemannian manifold whose sectional curvature is negative and bounded away from zero and the injectivity radius is positive. Let 0 W ! 1.M / be a homomorphism whose image is neither trivial nor infinite cyclic. Then the latter does not fix a point on the ideal boundary of the universal cover of M 0.
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